Uncertainty and predictability in geophysics: Chaos and multifractal insights

Geophysical Monograph Series

Volumn 150, Issue , 2004, Pages 317-334

  Schertzer, Daniel ^a   Lovejoy, Shaun ^b  

^a Universite Paris Est   (France)

^b MCGILL UNIVERSITY   (Canada)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84883058793     PISSN: 00658448     EISSN: 23288779     Source Type: Book Series    
DOI: 10.1029/150GM25     Document Type: Chapter

References (141)

1. Adelfang, S.I., On the relation between wind shears over various intervals, Journal of Atmospheric Sciences, 10, 138, 1971.
2. Ameodo, A., N. Descoter, and S.G. Roux, Intermittency, Log-Normal Statistics, and Multifractal Cascade Process in High-Reso-lution Satellite Images of Cloud Structure, Phys. Rev Letters, 83 (6), 1255-1258, 1999.
3. Arnold, L., Random Dynamical Systems, 586 pp. pp., Springer, Berlin Heidelberg New York, 1998.
4. Badii, R., and A. Politi, Pys. Rev. Lett., 52, 1661, 1984.
5. Bak, P., C. Tang, and K. Weiessenfeld, Self-Organized Criticality: An explanation of 1/f noise, Physical Review Letter, 59, 381-384, 1987.
6. Batchelor, G.K., The theory of homogeneous turbulence, Cambridge University Press, Cambridge, 1953.
7. Batchelor, G.K., and A.A. Townsend, The Nature of turbulent motion at large wavenumbers, Proceedings of the Royal Society of London, A 199, 238, 1949.
8. Benzi, R., G. Paladin, G. Parisi, and A. Vulpiani, On the multifractal nature of fully developped turbulence, Journal of Physics A, 17, 3521-3531, 1984.
9. Bolgiano, R., Turbulent spectra in a stably stratified atmosphere, J. Geophys. Res., 64, 2226, 1959.
10. Buizza, R., M.J. Miller, and T.N. Palmer, Stochastic Simlulation of Model Uncertainties in the ECWMF Ensemble Prediction System, Q. J. R. Meteorol. Soc., 125, 28887-2908, 1999.
11. Chamey, J.G., Geophys. Publ., 17,1, 1948.
12. Chamey, J.G., J. Atmos. Sci, 28, 1087, 1971.
13. Chamey, J.G., and et al., The feasibility of a global observation and analysis experiment, Bull. Amer. Meteor. Soc., 47,200-220,1966.
14. Chigirinskaya, Y., D. Schertzer, S. Lovejoy, A. Lazarev, and A. Ordanovich, Unified multifractal atmospheric dynamics tested in the tropics, part I: horizontal scaling and self organized criticality, Nonlinear Processes in Geophysics, 1 (2/3), 105-114, 1994.
15. Corrsin, S., On the spectrum of Isotropic Temperature Fluctuations in an isotropic Turbulence, Journal of Applied Physics, 22, 469-473, 1951.
16. Deidda, R., Rainfall downscaling in a space-time multifractal framework, WaterResour. Res., 36, 1779-1794, 2000.
17. Eckmann, J.P., and D. Ruelle, Ergodic theory of chaos and strange attractors, Rev. Mod. Phys., 57 (3), 617-656, 1985.
18. Ehrendorfer, M., The Liouville equation and its potential usefulness for the prediction of forecast skill. Part I: Theory, Mon. Wea. Rev., 122, 703-713, 1994.
19. Endlich, R.M., R.C. Singleton, and J.W. Kaufman, Spectral Analyes of detailed vertical wind profiles, Journal of Atmospheric Sciences, 26, 1030-1041, 1969.
20. Epstein, E.S., Stochastic dynamic prediction, Tellus, 21, 739-759, 1969.
21. Essex, C., Correlation dimension and data sample size, in Non-linear variability in geophysics: Scaling and Fractals, edited by D. Schertzer, and S. Lovejoy, pp. 93-98, Kluwer, Dordrecht, 1991.
22. Essex, C., T. Lookman, and M.A.H. Nerenberg, The climate attractor over short timescales, Nature, 326, 64, 1987.
23. Fjortoft, R., On the changes in the spectral distribution of kinetic energy in two dimensional, nondivergent flow, Tellus, 7, 168-176, 1953.
24. Forster, D., D.R. Nelson, and M.J. Stephen, Large distance and long time properties of a randomly stirred fluid, Phys. Rev. A, 16, 732-749, 1977.
25. Fraedrich, K., Estimating the dimensions of weather and climate attractors, J. Atmos. Sci., 43, 419, 1986.
26. Frisch, U., Turbulence: The Legacy of A. N. Kolmogorov, 296 pp., Cambridge University Press, Cambridge, 1995.
27. Frisch, U., M. Lesieur, and D. Schertzer, Comment on the QuasiNormal Markovian Approximation for Fully-Developed Turbulence, J. Fluid Mech., 97, 181-192, 1980.
28. Gage, K.S., Evidence for k'5/3 law inertial range in meso-scale two dimensional turbulence, Journal of Atmospheric Sciences, 36, 1979, 1979.
29. Gardiner, C.W., Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences., 442 pp. pp., Springer, Berlin, 1990.
30. Grassberger, P., Generalized dimensions of strange attractors, Physical Review Letter, A 97,221, 1983.
31. Grassberger, P., Do climatic attractors exist?, Nature, 322, 609-612, 1986.
32. Grassberger, P., and I. Procaccia, Measuring the strangeness of Strange atractors, Physica, 9D, 189-208, 1983.
33. Halsey, T.C., M.H. Jensen, L.P. Kadanoff, I. Procaccia, and B. Shraiman, Fractal measures and their singularities: the characterization of strange sets, Physical Review A, 33, 1141-1151, 1986.
34. Hasselmann, K., Stochastic climate models. Part I. Theory, Tellus, 28, 473-485, 1976.
35. Hense, A., On the possible existence of a strange attractor for the southern oscillations, Contr. to Atmos. Phys., 60, 1987, 1986.
36. Hentschel, H.G.E., and I. Procaccia, The infinite number of generalized dimensions of fractals and strange attractors, Physica D, 8, 435-444, 1983.
37. Herring, J.R., D. Schertzer, J.P. Chollet, M. Larchevéque, M. Lesieur, and G.R. Newman, A comparative assesment of spectral closures as applied to passive scalar diffusion., J. Fluid Mech, 124,411-437, 1982.
38. Hilbert, D., Sur les problèmes future des mathématiques, in Comptes Rendus du 2 ème Congrès Internationa des Mathématiciens, pp. 58-114,, Gauthier-Villars, Paris, 1902.
39. Houtekamer, P., L. Lefaivre, J. Derome, H. Ritchie, and M. H., A system simulation approach to ensemble prediction, Monthly Weather Review, 124, 1225-1242, 1996.
40. Jayawardena, A. W., and F. Lai, Analysis and prediction of chaos in rainfall and stream flow time series, J. Hydrol., 153, 23-52, 1994.
41. Kolmogorov, A.N., Local structure of turbulence in an incompressible liquid for very large Raynolds numbers, Proc. Acad. Sci. URSS, Geochem. Sect., 30, 299-303, 1941.
42. Kraichnan, R.H., A theory of turbulence dynamics, in Second Symposium on Naval Hydrodynamics, pp. 29-44, Office of Naval Research, Washington, DC, 1958.
43. Kraichnan, R.H., The structure of isotropic turbulence at very high Reynolds numbers, J. Fluid Mech., 5, 497-543, 1959.
44. Kraichnan, R.H., Inertial ranges in two-dimensional turbulence, Physics of Fluids, 10, 1417-1423, 1967.
45. Kraichnan, R.H., Instability in fully developped turbulence, Phys. Fluids, 13, 569-575, 1970.
46. Kraichnan, R.H., An almost-Markovian Galilean-invariant turbulence model, J. Fluid Mech., 47, 513-524, 1971.
47. Landau, L.D., and E.M. Lifshitz, Fluid Mechanics, Pergamon Press, Oxford, 1987.
48. Laplace, P.S., Théorie Analytique des Probabilités, Gauthier-Villars, Paris, 1886.
49. Lazarev, A., D. Schertzer, S. Lovejoy, and Y. Chigirinskaya, Unified multifractal atmospheric dynamics tested in the tropics: part II, vertical scaling and Generalized Scale Invariance, Nonlinear Processes in Geophysics, 1, 115-123, 1994.
50. Leith, C.E., Atmospheric predictability and two-dimensional turbulence, J. Atmos. Sci., 28, 145-161, 1971.
51. Leith, C.E., and R.H. Kraichnan, Predictability of turbulent flows, J. Atmos. Sci, 29, 1041-1058, 1972.
52. Leslie, D.C., Developments in the theory of turbulence, 368 pp., Clarendon Press, Oxford, 1973.
53. Lilley, M., K. Strawbridge, S. Lovejoy, and D. Schertzer, 23/9 dimensional anisotropic scaling of passive admixtures using lidar data of aerosols, Phys. Rev. E (in press), 2004.
54. Lilly, D., and E.L. Paterson, Aircraft measurements of atmospheric kinetic energy spectra, Tellus, 35A, 379-382, 1983.
55. Lilly, D.K., Dynamics of rotating thunderstorms, in Meso-scale Meteorology -Theories, Observations and models, edited by D.K. Lilly, and T. Galchen, D. Reidel, Dordrecht, 1983.
56. Lilly, D.K., Theoretical predictability of small-scale motions, in Turbulence and predictability in geophysical fluid dynamics and climate dynamics, edited by M. Ghil, R. Benzi, and G. Parisi, pp. 281-280, North Holland, Amsterdam, 1985
57. Liouville, J., Sur la Théorie de la Variation des constantes arbitraires, Journal des Mathématiques Pures et Appliquées, 3, 342-349, 1838.
58. Lorenz, E.N., Deterministic nonpereodic flow, J. Atmos. Sci., 20, 130-141, 1963a.
59. Lorenz, E.N., The predictability of hydrodynamic flow, Trans. New York Acad. Sc., Ser. 2 (25), 409-432, 1963b.
60. Lorenz, E.N., The predictability of a flow which possesses many scales of motion, Tellus, 21, 289-307, 1969.
61. Lorenz, E.N., Atmospheric predictability experiments with a large numerical model, Tellus, 34 (505-513), 1982.
62. Lovejoy, S., S. Pecknold, and D. Schertzer, Stratified multifractal magnetization and surface geomagnetic fields -1 Spectral analysis and modelling, Geophys. Inter. J., 144, 1-22, 2001.
63. Lovejoy, S., D. Schertzer, and J.D. Stanway, Direct Evidence of Multifractal Atmospheric Cascades from Planetary Scales down to 1 km., Phys. Rev. Letter, 86 (22), 5200-5203, 2001.
64. Lovejoy, S., D. Schertzer, and A.F. Tuck, Fractal Aircraft Trajectories and anomalous turbulent statistics, Phys.Rev. E (in press), 2004.
65. Lumley, J., Journal of the Atmospheric Sciences, 21, 99, 1963.
66. Lyapunov, M.A., Probleme général de la stabilité du mouvement, Annales Fac. Siences Toulouse, 9, 1907.
67. Mandelbrot, B., Random multifractals: negative dimensions and the resulting limitations of the thermodynamic formalism, in Turbulence and Stochastic Processes, edited by J.C.R.H.e. al., The Royal Society, London, 1991.
68. Marsan, D., Multifractals espace-temps, dynamique etprédicibilité; application aux precipitations, Docteur de 1’Université thesis, Universite Paris 6, Paris, 1998.
69. Marsan, D., D. Schertzer, and S. Lovejoy, Causal Space-Time Multifractal modelling of rain, J. Geophy. Res., D 31 (26), 26,333-26346, 1996.
70. Marzocchi W., F. Mulargia, and G. Gonzato, Detecting low-dimensional chaos in geophysical time series, J. Geophys. Res., 102, 3195-3209, 1997.
71. May, R.M., Simple mathematical models with very complicated dynamics, Nature, 261, 459-467, 1976.
72. Meneveau, C., and K.R. Sreenivasan, Simple multifractal cascade model for fully develop turbulence, Physical Review Letter, 59 (13), 1424-1427, 1987.
73. Métais, O., and M. Lesieur, Statistical predictability of decaying turbulence, J. Atmos. Sci., 43, 857-870, 1986.
74. Millionshchikov, M.D., Theory of homogeneous isotropic turbulence, Izvestiya, Ser. Geogr. Geophys., 5 (433-446), 1941.
75. Molteni, R.R., R. Buizza, and T.N. Palmer, The ECMWF ensemble prediction system: methodology and validation, Q. J. R. Meteorol. Soc.,122, 73-119, 1996.
76. Monin, A.S., Weather forecasting as a problem in physics, MIT press, Boston Ma, 1972.
77. Nastrom, G.D., and K.S. Gage, A first look at wave number spectra from GASP data, Tellus, 35, 383, 1983a.
78. Nastrom, G.D., and K.S. Gage, A first look at wavenumber spectra from GASP data, Tellus, 1983b.
79. Naud, C., Schertzer, D., and S. Lovejoy, Fractional Integration and radiative transfer in multifractal atmospheres, in Stochastic Models in Geosystems, edited by W. Woyczynski, and S. Molchansov, pp. 239-267, Springer-Verlag, 1996.
80. Nerenberg, M.A.H., and E. C, Correlation dimension and systematic geometric effects, Phys. Rev. A, 42 (12), 7065-7074, 1990.
81. Nicolis, C., and G. Nicolis, Is there a climate attractor?, Nature, 311, 529-532, 1984.
82. Nicolis, G., Introduction to Nonlinear Science, Cambridge Uni-veristy Press, Cambridge, 1995.
83. Obukhov, A., Structure of the temperature field in a turbulent flow, Izv. Akad. Nauk. SSSR. Ser. Geogr. I Geofiz, 13, 55-69, 1949.
84. Obukhov, A.M., On the distribution of energy in the spectrum of turbulent flow, Izvestiya, Geogr. Geophys., 5, 453-466, 1941.
85. Obukhov, A.N., Effect of Archimedian forces on the structure of the temperature field in a temperature flow, Sov. Phys. Dokl., 125, 1246, 1959.
86. Oono, Y., Progr. theor. phys. Suppl., 99, 165, 1989.
87. Orszag, S.A., Analytical Theories of Turbulence, J. Fluid. Mech., 41, 362-386, 1970.
88. Osborne, A.R., and A. Provenzale, Finite correlation dimension for stochastic systems with power-law spectra, Physica D, 35,357-381, 1989.
89. Oseledets, V.I., A multiplicative ergodic theorem.lyapunov characteristic numbers for dynamical systems, Trans. Mocscow Maths Soc., 19, 197-231, 1968.
90. Over, T.M., and V.J. Gupta, A space time theory of mesoscale rainfall using random cascades, J Geophys. Res., D 31, 26,319-26,331, 1996.
91. Packard, N.H., J.P. Crutchfield, D. Farmer, and R.S. Shaw, Geometry from a time series, Phys. Rev. Lett., 45 (9), 712-716., 1980.
92. Palmer, T.M., Predicting uncertainty in forecasts of weather and climate, Reports on Progress in Physics, 63, 71-116, 2000.
93. Palmer, T.N., Predicting uncertainty in forecast of weather and climate, ECWMF, Reading, U.K., 1999.
94. Panofsky, H.A., and I. Van der Hoven, Quarterly J. of the Royal Meteorol. Soc. (81), 603, 1955.
95. Parisi, G., and U. Frisch, On the singularity structure of fully developed turbulence, in Turbulence and predictability in geophysical fluid dynamics and climate dynamics, edited by M. Ghil, R. Benzi, and G. Parisi, pp. 84-88, North Holland, Amsterdam, 1985.
96. Pecknold, S., S. Lovejoy, and D. Schertzer, The morphology and texture of anisotropic multifractals using generalized scale invariance, in Stochastic Models in Geosystems, edited by W. Woyinski, pp. 269-311, Inst, for Math, and its Appl., 1996.
97. Pecknold, S., S. Lovejoy, D. Schertzer, and C. Hooge, Multifractals and the resolution dependence of remotely sensed data: Generalized Scale Invariance and Geographical Information Systems, in Scaling in Remote Sensing and Geographical Information Systems, edited by M.G. D. Quattrochi, pp. 361-394, Lewis, Boca Raton, Florida, 1997.
98. Pecknold, S., S. Lovejoy, D. Schertzer, C. Hooge, and J.F. Malouin, The simulation of universal multifractals, in Cellular Automata: prospects in astronomy and astrophysics, edited by J.M. Perdang, and A. Lejeune, pp. 228-267, World Scientific, 1993.
99. Pedlosky, J., Geophysical fluid Dynamics, Springer-Verlag, Berlin, Heidelberg, New York, 1979.
100. Pietronero, L., and A.P. Siebesma, Self-similarity of fluctuations in random multiplicative processes, Physical Review Letter, 57,1098,1986.
101. Pinus, N.Z., The energy of atmospheric macro-turbulence, Izvestiya, Atmospheric and Oceanic Physics, 4, 461, 1968.
102. Richardson, L.F., Weather prediction by numerical process, Cambridge University Press republished by Dover, 1965, 1922.
103. Richardson, L.F., Atmospheric diffusion shown on a distance-neighbour graph, Proc. Roy. Soc., A110,, 709-737, 1926.
104. Robert, R., and C. Rosier, Long range predictability of atmospheric flows, Nonlinear Processes in Geophysics, 8, 55-67, 2001.
105. Robinson, G.D., Some current projects for global meteorological observation and experiment, Quart. J. Roy. Meteor. Soc., 93, 409-418, 1967.
106. Robinson, G.D., The predictability of a dissipative flow, Quart. J. Roy. Meteor. Soc., 97, 300-312, 1971.
107. Rodriguez-Iturbe, I., B. Febres de Power, and J.B. Valdés, Chaos in rainfall, Water Resources Research, 25, 1667-1675, 1989.
108. Ruelle, D., Characteristic exponents and invariant manifolds in Hilbert space, Annals of Mathematics, 115, 243-290, 1982.
109. Ruelle, D., Proc. R. Soc.,A427, 241-248, 1990.
110. Sauer, T., J. York, and M. Casdagli, Embedology, J. Stat. Phys, 65, 579, 1991.
111. Schertzer, D., M. Larcheveque, J. Duan, V.V. Yanovsky, and S. Lovejoy, Fractional Fokker-Planck equation for nonlinear stochastic differential equations driven by non-Gaussian Lévy stable noises, J. Math. Phys, 41 (1), 200-212, 2001.
112. Schertzer, D., and S. Lovejoy, On the Dimension of Atmospheric motions, in Turbulence and Chaotic phenomena in Fluids, edited by T. Tatsumi, pp. 505-512, Elsevier Science Publishers B. V., Amsterdam, 1984.
113. Schertzer, D., and S. Lovejoy, The dimension and intermittency of atmospheric dynamics, in Turbulent Shear Flow 4, edited by B. Launder, pp. 7-33, Springer-Verlag, 1985a.
114. Schertzer, D., and S. Lovejoy, Generalised scale invariance in turbulent phenomena, Physico-Chemical Hydrodynamics Journal, 6, 623-635, 1985b.
115. Schertzer, D., and S. Lovejoy, Physical modeling and Analysis of Rain and Clouds by Anisotropic Scaling of Multiplicative Processes, Journal of Geophysical Research, D 8 (8), 9693-9714, 1987.
116. Schertzer, D., and S. Lovejoy, Hard and Soft Multifractal processes, Physica A, 185, 187-194, 1992.
117. Schertzer, D., and S. Lovejoy, Lecture Notes: Nonlinear Variability in Geophysics 3: Scaling and Mulitfractal Processes in Geophysics, 292 pp., Institut d’Etudes Scientifique de Cargèse, Cargèse, France, 1993.
118. Schertzer, D., and S. Lovejoy, From scalar to Lie cascades: joint multifractal analysis of rain and clouds processes, in Space/Time Variability and Interdependance of Hydological Processes, edited by R.A. Feddes, pp. 153-173, University Press, Cambridge, 1995.
119. Schertzer, D., and S. Lovejoy, Universal Multifractals do Exist!, J. Appl. Meteor., 36, 1296-1303, 1997
120. Schertzer, D., and S. Lovejoy, Space-time Complexity and Multifractal Predictability, Physica A, 38 (1-2), 173-186, 2004.
121. Schertzer, D., S. Lovejoy, and P. Hubert, An Introduction to Stochastic Multifractal Fields, in ISFMA Symposium on Environmental Science and Engineering with related Mathematical Problems, edited by A. Em, and W. Liu, pp. 106-179, High Education Press, Beijing, 2002a.
122. Schertzer, D., S. Lovejoy, F. Schmitt, I. Tchiguirinskaia, and D. Marsan, Multifractal cascade dynamics and turbulent intermittency, Fractals, 5 (3), 427-471, 1997.
123. Schertzer, D., I. Tchiguirinskaia, S. Lovejoy, P. Hubert, and H. Bend-joudi, Which chaos in the rain-runoff process?, J. Hydrological Sciences, 47(1), 139-148, 2002b.
124. Schuster, H.G., Deterministic Chaos, VCH, New York, 1988.
125. Sedov, L., Similitudes et Dimensions en Mécanique, MIR, Moscow, 1972.
126. Shur, G., Trudy, 43 (79), 1962.
127. Takens, Detecting strange attractors in turbulence, in Dynamical Systems and Turbulence, edited by D.A. Rand, and L.S. Young, pp. 366-381, Springer-Verlag, Berlin, 1980.
128. Tchiguirinskaia, I., Scale invariance and stratification: the unified multifractal model of hydraulic conductivity, Fractals, 10 (3), 329-334, 2002.
129. Theiler, J., Some comments on the correlation dimension of a 1/f-alpha noise, Phys.Lett., A (155), 480, 1991.
130. Thompson, P.D., Uncertainty of intial state as a factor in the predictability of large scale atmospheric flow patterns, Tellus, 9, 275-295, 1957.
131. Toth, Z., and E. Kalnay, Ensemble Forecasting at NMC: the generation of perturbations, Bull. Amer. Meteor. Soc., 74, 2317-2330, 1993.
132. Tsonis, A.A., Chaos: Frome Theory to Application, 274 pp., Plenum, New York, 1992.
133. Tsonis, A.A., J.B. Eisner, and K.P. Georgakakos, Estimating the dimension of weather and climate attractors: important issues about the procedure and interpretation, J. Atm. Sci., 50,2549-255,1993.
134. Van der Hoven, I., Power spectrum of horizontal wind speed in the frequency range from.0007 to 900 cycles per hour, Journal of Meteorology, 14, 160-164, 1957.
135. Van Zandt, T.E., A universal spectrum of buoyancy waves in the atmosphere, Geophysical Research Letter, 9, 575-578, 1982.
136. Vinnichenko, N.K., The kinetic energy spectrum in the free atmosphere for 1 second to 5 years, Tellus, 22, 158, 1969.
137. Weinstock, J., J. Atmos. Sci., 35, 634, 1978.
138. Wiin-Nielsen, A., On the annual variation and spectral distribution of atmospheric energy, Tellus, 19, 540-559, 1967.
139. Wilson, J., D. Schertzer, and S. Lovejoy, Physically based modelling by multiplicative cascade processes, in Non-linear variability in geophysics: Scaling and Fractals, edited by D. Schertzer, and S. Lovejoy, pp. 185-208, Kluwer, Dordrecht, 1991.
140. Yaglom, A.M., The influence on the fluctuation in energy dissipation on the shape of turbulent characteristics in the inertial interval, Sov. Phys. Dokl., 2, 26-30, 1966.
141. Yaglom, A.M., A. N. Kolmogorov as a fluid mechanician and founder of a school in turbulence research, Ann. Rev. Fluid Mech., 26, 1-22, 1994.